(c) Let D = R² \ {(0,0)} and consider the vector field defined on D defined as: X -Y x² + y² x² + y² (i) Show that g is conservative on D. g(x, y) = i+ $.9 C (ii) Let C be the unit circle around the origin. Suppose a fellow student proposes the following statement "Since C is closed and g is conservative, we must have: 9.ds = 0 j since integrals of conservative vector fields around closed loops are always zero.' 29 By calculating g.ds directly, show that your fellow student's state- ment is not true and explain why they were incorrect in their reasoning. [9 (iii) Let a and ß be two arbitrary positive real numbers and let y be an arbitrary smooth curve starting at (–a, 0), ending at (3,0) that only traverses the lower half of the xy-plane. Evaluate the integral: ds
(c) Let D = R² \ {(0,0)} and consider the vector field defined on D defined as: X -Y x² + y² x² + y² (i) Show that g is conservative on D. g(x, y) = i+ $.9 C (ii) Let C be the unit circle around the origin. Suppose a fellow student proposes the following statement "Since C is closed and g is conservative, we must have: 9.ds = 0 j since integrals of conservative vector fields around closed loops are always zero.' 29 By calculating g.ds directly, show that your fellow student's state- ment is not true and explain why they were incorrect in their reasoning. [9 (iii) Let a and ß be two arbitrary positive real numbers and let y be an arbitrary smooth curve starting at (–a, 0), ending at (3,0) that only traverses the lower half of the xy-plane. Evaluate the integral: ds
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(c) Let D = R² \ {(0,0)} and consider the vector field defined on D defined as:
-Y
X
i+
x² + y² x² + y²
j
g(x, y)
=
(i) Show that g is conservative on D.
(ii)
Let C be the unit circle around the origin. Suppose a fellow student
proposes the following statement
"Since C is closed and g is conservative, we must have:
$
g.ds=0
since integrals of conservative vector fields around closed loops
are always zero.
99
By calculating g.ds directly, show that your fellow student's state-
ment is not true and explain why they were incorrect in their reasoning.
(iii) Let a and 3 be two arbitrary positive real numbers and let y be an
arbitrary smooth curve starting at (-a, 0), ending at (3, 0) that only
traverses the lower half of the xy-plane. Evaluate the integral:
[₁
9.ds
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i have seen that you have done part i of the question. i would like to ask if you can do part ii) and part iii) as well, since i dont see any solutions for those questions. i will add the image for the question again.
Transcribed Image Text:(c) Let D = R² \ {(0,0)} and consider the vector field defined on D defined as:
-Y
X
i+
x² + y² x² + y²
j
g(x, y)
=
(i) Show that g is conservative on D.
(ii)
Let C be the unit circle around the origin. Suppose a fellow student
proposes the following statement
"Since C is closed and g is conservative, we must have:
$
g.ds=0
since integrals of conservative vector fields around closed loops
are always zero.
99
By calculating g.ds directly, show that your fellow student's state-
ment is not true and explain why they were incorrect in their reasoning.
(iii) Let a and 3 be two arbitrary positive real numbers and let y be an
arbitrary smooth curve starting at (-a, 0), ending at (3, 0) that only
traverses the lower half of the xy-plane. Evaluate the integral:
[₁
9.ds
Solution
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